Conduction, heat integral equations

In writing this equation, it has been noted that since be lies in the freestream where the temperature is constant, there can be no heat transfer into the control volume through it. Longitudinal conduction effects have also been ignored because the boundary layer is assumed to be thin. This is consistent with the neglect of the effects of longitudinal viscous forces in the derivation of the momentum integral equation. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

Assuming isothermal plate and constant fluid properties and negligible conduction heat transfer in the x-direction, the integral x-momentum equation for the hydrodynamic boundary layer is... 

Equation 3.43g compares the timescale for radial heat dispersion in the solid phase with the one for internal heat conduction. For catalysts with good heat conduction properties and low particle-to-bed diameter ratios, A l. In this case, the surface boimdary condition is homogeneous and of Robin type, as given by the first terms on each side of (3.42b). A similar dimensionless number related with dispersion in the axial direction also appears, but its magnitude is considered much smaller than that of the other parameters in Equation 3.43, due to the geometrical reasons explained earlier. Note that Equations 3.32 and 3.34 are obtained by integrating Equation 3.41 with respect to over the pellet domain and using Equation 3.42 as boundary conditions. 

In the coupled BIE/FD formulation mentioned above, the requirement for the evaluation of the domain integral is certainly a costly feature. Zheng and Phan-Thien (1992), who considered a boundary integral equation for the following problem of heat conduction with a heat source, have employed an alternative approach that transforms the domain integral to a boundary integral. 

The Laplace-transform based boundary integral equation was proposed by Rizzo and Shippy (1970) to solve transient heat conduction problems. 

As has been mentioned in Sect. 7.3, the continuous injection-molding operation results in a cyclic heat transfer behavior in the mold, after a short transient period. The cycle-averaged temperature can be represented by a steady state heat conduction equation, i.e., Eq. 7.10. The mold cooling analysis can be greatly simplihed by solving the steady state problem. The boundary integral equation of ( 7.10) is... 

Equation (6.109) is known as Fourier s law of heat conduction. As noted above, there will be a potential contribution as well. Solution to the integral equation, Eq. (6.95), for the function A w) gives numerical values of the thermal conductivity from Eq. (6.110)." ... 

Coen et al (1987) report an approach, new to electrochemistry, to reduce the computation time required for a 2-D system (the current at a micro-band electrode). The diffusion equation is Laplace transformed, converted to an integral equation and solved, still in Laplace space, for the concentration gradient at the boundaries. They developed an efficient algorithm for the inverse Laplace transformation, which then yields the current as a function of time. Clearly, this is not for everyone at least one of the authors is a mathematician. The method has been used previously (Rizzo and Shippy, 1970) to simulate heat conduction. 

On the basis of the theory of numerical methods and mathematical modeling the problem of the calculation and forecast of the distribution of the temperature field in a two-phase nanocomposite environment is solved. The mathematical statement of the problem is formulated as the integral equation of thermal balance with a heat flux taken into account, which changes according to Fourier s law. Jumps of enthalpy and heat conductivity coefficient are considered. Various numerical schemes and methods are examined and the best one is selected - the method of control volume. Calculation of the dynamics of the temperature field in the nanostructure is hold using the software. 

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

Heat conductivity has been studied by placing the end particles in contact with two thermal reservoirs at different temperatures (see (Casati et al, 2005) for details)and then integrating the equations of motion. Numerical results (Casati et al, 2005) demonstrated that, in the small uj regime, the heat conductivity is system size dependent, while at large uj, when the system becomes almost fully chaotic, the heat conductivity becomes independent of the system size (if the size is large enough). This means that Fourier law is obeyed in the chaotic regime. 

The first term on the right-hand side of Eq. (3.8) corresponds to a two-dimensional heat flux. The second part has, in this model, the physical interpretation of a local heat sink or source perpendicular to the x-y-plane. The equation can be further simplified by introducing an integral heat conductivity within the membrane ... 

The equations for the diffusion profile can be obtained from the heat-conduction equations of Carslaw and Jaeger [Ref. 3, Eq. 9.4 (10)] by using the substitutions we have indicated. The subject is discussed from a different approach by Adams, Quan, and Balkwell (I). The profiles can then be integrated over the volume of the sphere to obtain the uptake as a function of time. 

The quantity, h, in Equation 5 is not likely to be greatly different from its value in a plane adiabatic combustion wave. Taking x as the coordinate normal to such wave, h becomes the integral of the excess enthalpy per unit volume along the x-axis, so that the differential quotient, dh/dx, represents the excess enthalpy per unit volume in any layer, dx. Assuming the layer to be fixed with respect to a reference point on the x-axis, the mass flow passes through the layer in the direction from the unbumed, w, to the burned, 6, side at a velocity, S, transporting enthalpy at the rate Sdh/dx. Because the wave is in the steady state, heat flows by conduction at the same rate in the opposite direction, so that... 

Integration of this equation can be readily obtained for the case where the heat conduction term can be neglected (adiabatic approximation). This is not too bad an approximation, as T — T0 is small during the induction period. With this approximation in mind, Equation 23 becomes... 

Lotkin (L10) gives a scheme for numerical integration of the heat conduction equation in a finite ablating slab, using unequal subdivisions in both space and time variables. Near the melting surface it is advantageous to choose rather small integration steps. Stability characteristics of the method are established. 

Taking account of the boundary conditions, this equation can be integrated by elementary methods at each given instant and in a given layer. This determines the function late stage the plane field may be represented in the form H2 = curl (n ), where n = (0,0,1). After this the function (which now coincides with the vector potential component Az) is also subject to an equation of the heat conduction type. Consequently, H2 decays asymptotically. 

The first type, which includes, for example, the problem of strong explosion or propagation of heat in a medium with nonlinear thermal conductivity [3], is characterized by the fact that the exponents are found from physical considerations, from the conservation laws and their dimensionality. In addition, the exponents turn out to be rational numbers. The task of the calculation is to find the dimensionless functions by integration of ordinary differential equations. After this the problem is completely solved, since the numerical constants are determined by normalizing the solution to the conserved quantity (the total energy released in these examples). 

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